Question: Solve for $z$, $ \dfrac{8}{z - 4} = \dfrac{3z + 6}{3z - 12} - \dfrac{10}{5z - 20} $
Explanation: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $z - 4$ $3z - 12$ and $5z - 20$ The common denominator is $15z - 60$ To get $15z - 60$ in the denominator of the first term, multiply it by $\frac{15}{15}$ $ \dfrac{8}{z - 4} \times \dfrac{15}{15} = \dfrac{120}{15z - 60} $ To get $15z - 60$ in the denominator of the second term, multiply it by $\frac{5}{5}$ $ \dfrac{3z + 6}{3z - 12} \times \dfrac{5}{5} = \dfrac{15z + 30}{15z - 60} $ To get $15z - 60$ in the denominator of the third term, multiply it by $\frac{3}{3}$ $ -\dfrac{10}{5z - 20} \times \dfrac{3}{3} = -\dfrac{30}{15z - 60} $ This give us: $ \dfrac{120}{15z - 60} = \dfrac{15z + 30}{15z - 60} - \dfrac{30}{15z - 60} $ If we multiply both sides of the equation by $15z - 60$ , we get: $ 120 = 15z + 30 - 30$ $ 120 = 15z$ $ 120 = 15z $ $ z = 8$